Stress collisional

The simplest theory of impact, known as stereomechanics, deals with the impact between rigid bodies using the impulse-momentum law. This approach yields a quick estimation of the velocity after collision and the corresponding kinetic energy loss. However, it does not yield transient stresses, collisional forces, impact duration, or collisional deformation of the colliding objects. Because of its simplicity, the stereomechanical impact theory has been extensively used in the treatment of collisional contributions in the particle momentum equations and in the particle velocity boundary conditions in connection with the computation of gas-solid flows. 

In this chapter we shall first outline the basic concepts of the various mechanisms for energy redistribution, followed by a very brief overview of collisional intennoleciilar energy transfer in chemical reaction systems. The main part of this chapter deals with true intramolecular energy transfer in polyatomic molecules, which is a topic of particular current importance. Stress is placed on basic ideas and concepts. It is not the aim of this chapter to review in detail the vast literature on this topic we refer to some of the key reviews and books [U, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, and 32] and the literature cited therein. These cover a variety of aspects of tire topic and fiirther, more detailed references will be given tliroiighoiit this review. We should mention here the energy transfer processes, which are of fiindamental importance but are beyond the scope of this review, such as electronic energy transfer by mechanisms of the Forster type [33, 34] and related processes. 

Regardless of these short-ranged cohesive forces, the formation and stability of particle clusters in a fluidized bed appears to be a multistep process [27], Some shear (as in two particles grazing each other) may be needed to promote collisional cooling, but less than that perhaps in the dense emnlsion of a fluidized bed. Perhaps the lower particle concentration in a babble provides the environment where clnster stability is promoted for the smaller particles. Collisional stresses in the emnlsion may be too high and the cohesive forces may be too low to have long-lasting particle clusters. Indeed, the only evidence of particle clnsters in fluidized beds offered here is that the clusters are located near the bubbles. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

The averaged collisional stress between a particle and a group of neighboring particles of the same diameter in a shear flow can be expressed by —(/ipAp), where /Lip is given by... 

For a collision-dominated dense suspension, the collisional stress tensor Pc is given by... 

Fa Carried mass force vector Pc Collisional stress tensor... 

Hence, the collisional rate of dissipation and the collisional stress tensor are related to the coefficient of restitution by Eq. (5.284) and Eq. (5.277). 

From the momentum equation and the constitutive relation of collisional stress tensor, show that Tc is governed by... 

Pc Collisional stress tensor of particles Ui Gas velocity component in direction i... 

Brittle erosion is the loss of material from a solid surface due to fatigue cracking and brittle cracking caused by the normal collisional force Fn. Materials with very limited capacity for elastic and plastic deformation, such as ceramics and glass, respond to particle impacts by fracturing. The yield stress for brittle failure Fb for normal impacts is about... 

Das and Bhattacharjee236 derive the frequency and shear dependent viscosity of a simple fluid at the critical point and find good agreement with recent experimental measurements of Berg et al.237 Ernst238 calculates universal power law tails for single and multi-particle time correlation functions and finds that the collisional transfer component of the stress autocorrelation function in a classical dense fluid has the same long-time behaviour as the velocity autocorrelation function for the Lorentz gas, i.e. 

Ding and Gidaspow [16], for example, derived a two-phase flow model starting with the Boltzmann equation for the distribution function of particles and incorporated fluid-particle interactions into the macroscopic equations. The governing equations were derived using the classical concepts of kinetic theory. However, to determine the constitutive equations they used the ad hoc distribution functions proposed by Savage and Jeffery [65]. The resulting macroscopic equations contain both kinetic - and collisional pressures but only the collisional deviatoric stresses. The model is thus primarily intended for dense particle flows. 

Srivastava and Sundaresan [127] calculated the total stress as a linear sum of the kinetic, collisional, and frictional stress components, where each of the contributions are evaluated as if they were alone. The extended particle pressure and viscosity properties are calculated as ... 

There is one additional mechanism for momentum transfer which does not seem to have been mentioned in the polymer literature. A bead of one dumbbell may collide with a bead of another dumbbell in such a way that their centers at the time of collision are on opposite sides of the plane at which the stress is being calculated. Such collisions would result in an instantaneous transfer of momentum between the two dumbbell centers and thus contribute to the stress at the plane. For very dilute solutions the contribution of this collisional momentum transfer would clearly be much smaller than those associated with mechanisms (i) and (ii) above. 

This particulate stress represents the kinetic components of granular momentum transfer, and includes both the viscous contribution due to the small-scale random motion of individual particles as well as the macroscopic turbulence contributions due to collective random motions such as eddies and bubbles (Sun, Chen, and Chao, 1990). The complete granular stress should consist of this particulate stress component and a collisional stress component. 

In this chapter we shall first outline the basic concepts of the various mechanisms for energy redistribution, followed by a very brief overview of collisional intermolecular energy transfer in chemical reaction systems. The main part of this chapter deals with true intramolecular energy transfer in polyatomic molecules, which is a topic of particular current importance. Stress is placed on basic ideas and concepts. It is not the aim of this chapter to review in detail the vast literature on this topic we refer to some of the key reviews and books [JT,... 

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